The subsequent problem is to see how the properties of \(\psi \)  impact the robustness and efficiency of \(T(X_1^n)\)  when estimating \(T(P)\)  or \(\mathbb {E}[X]\) . From robust statistics theory it is known that \(\psi \)  is strongly related to the influence function of the associated M-estimator.
The influence function is a classical tool used to quantify the robustness of an estimator, see for example [1]}, [2]}, [3]}, [4]} in which are derived properties such as the asymptotic variance or the breakdown point of the estimator \(T(X_1^n)\)  using the influence function.
The influence function is the Gâteaux derivative of \(T\)  evaluated in the Dirac distribution in a point \(x\in \mathbb {R}^d\)  and in the case of M-estimators, from [2]}, the influence function takes the following simple form:
\(\mathrm {IF}(x,T,P)=M_{P,T}^{-1}\frac{x-T(P)}{\Vert x-T(P)\Vert }\psi (\Vert x-T(P)\Vert ),\) 
